# Properties

 Label 507.2.k.a Level $507$ Weight $2$ Character orbit 507.k Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.k (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -1 - \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -1 - \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 + 4 \zeta_{12}^{2} ) q^{12} + 4 \zeta_{12}^{2} q^{16} + ( -3 + 3 \zeta_{12} + 5 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{19} + ( -5 + \zeta_{12} + \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{21} -5 \zeta_{12}^{3} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -6 - 2 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{28} + ( 6 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{31} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{36} + ( -4 + 7 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{37} + ( 1 + \zeta_{12}^{2} ) q^{43} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{48} + ( -6 + 7 \zeta_{12} + 3 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{49} + ( -7 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{57} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{61} + ( 9 - 6 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( 7 - 2 \zeta_{12} - 9 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{67} + ( 8 + \zeta_{12} + \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{73} + ( 10 - 5 \zeta_{12}^{2} ) q^{75} + ( -4 - 6 \zeta_{12} + 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{76} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 10 - 10 \zeta_{12} - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{84} + ( 7 + 11 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{93} + ( -3 + 3 \zeta_{12} - 8 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{7} - 6q^{9} + O(q^{10})$$ $$4q - 8q^{7} - 6q^{9} + 8q^{16} - 2q^{19} - 18q^{21} - 16q^{28} + 14q^{31} - 22q^{37} + 6q^{43} - 18q^{49} - 12q^{57} + 30q^{63} + 10q^{67} + 34q^{73} + 30q^{75} + 4q^{76} - 18q^{81} + 24q^{84} + 36q^{93} - 28q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0.866025 + 1.50000i 1.73205 + 1.00000i 0 0 −2.86603 + 0.767949i 0 −1.50000 + 2.59808i 0
89.1 0 −0.866025 1.50000i −1.73205 1.00000i 0 0 −1.13397 4.23205i 0 −1.50000 + 2.59808i 0
188.1 0 −0.866025 + 1.50000i −1.73205 + 1.00000i 0 0 −1.13397 + 4.23205i 0 −1.50000 2.59808i 0
488.1 0 0.866025 1.50000i 1.73205 1.00000i 0 0 −2.86603 0.767949i 0 −1.50000 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.f odd 12 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.a 4
3.b odd 2 1 CM 507.2.k.a 4
13.b even 2 1 507.2.k.b 4
13.c even 3 1 507.2.f.b 4
13.c even 3 1 507.2.k.c 4
13.d odd 4 1 39.2.k.a 4
13.d odd 4 1 507.2.k.c 4
13.e even 6 1 39.2.k.a 4
13.e even 6 1 507.2.f.c 4
13.f odd 12 1 507.2.f.b 4
13.f odd 12 1 507.2.f.c 4
13.f odd 12 1 inner 507.2.k.a 4
13.f odd 12 1 507.2.k.b 4
39.d odd 2 1 507.2.k.b 4
39.f even 4 1 39.2.k.a 4
39.f even 4 1 507.2.k.c 4
39.h odd 6 1 39.2.k.a 4
39.h odd 6 1 507.2.f.c 4
39.i odd 6 1 507.2.f.b 4
39.i odd 6 1 507.2.k.c 4
39.k even 12 1 507.2.f.b 4
39.k even 12 1 507.2.f.c 4
39.k even 12 1 inner 507.2.k.a 4
39.k even 12 1 507.2.k.b 4
52.f even 4 1 624.2.cn.b 4
52.i odd 6 1 624.2.cn.b 4
65.f even 4 1 975.2.bp.a 4
65.g odd 4 1 975.2.bo.c 4
65.k even 4 1 975.2.bp.d 4
65.l even 6 1 975.2.bo.c 4
65.r odd 12 1 975.2.bp.a 4
65.r odd 12 1 975.2.bp.d 4
156.l odd 4 1 624.2.cn.b 4
156.r even 6 1 624.2.cn.b 4
195.j odd 4 1 975.2.bp.d 4
195.n even 4 1 975.2.bo.c 4
195.u odd 4 1 975.2.bp.a 4
195.y odd 6 1 975.2.bo.c 4
195.bf even 12 1 975.2.bp.a 4
195.bf even 12 1 975.2.bp.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 13.d odd 4 1
39.2.k.a 4 13.e even 6 1
39.2.k.a 4 39.f even 4 1
39.2.k.a 4 39.h odd 6 1
507.2.f.b 4 13.c even 3 1
507.2.f.b 4 13.f odd 12 1
507.2.f.b 4 39.i odd 6 1
507.2.f.b 4 39.k even 12 1
507.2.f.c 4 13.e even 6 1
507.2.f.c 4 13.f odd 12 1
507.2.f.c 4 39.h odd 6 1
507.2.f.c 4 39.k even 12 1
507.2.k.a 4 1.a even 1 1 trivial
507.2.k.a 4 3.b odd 2 1 CM
507.2.k.a 4 13.f odd 12 1 inner
507.2.k.a 4 39.k even 12 1 inner
507.2.k.b 4 13.b even 2 1
507.2.k.b 4 13.f odd 12 1
507.2.k.b 4 39.d odd 2 1
507.2.k.b 4 39.k even 12 1
507.2.k.c 4 13.c even 3 1
507.2.k.c 4 13.d odd 4 1
507.2.k.c 4 39.f even 4 1
507.2.k.c 4 39.i odd 6 1
624.2.cn.b 4 52.f even 4 1
624.2.cn.b 4 52.i odd 6 1
624.2.cn.b 4 156.l odd 4 1
624.2.cn.b 4 156.r even 6 1
975.2.bo.c 4 65.g odd 4 1
975.2.bo.c 4 65.l even 6 1
975.2.bo.c 4 195.n even 4 1
975.2.bo.c 4 195.y odd 6 1
975.2.bp.a 4 65.f even 4 1
975.2.bp.a 4 65.r odd 12 1
975.2.bp.a 4 195.u odd 4 1
975.2.bp.a 4 195.bf even 12 1
975.2.bp.d 4 65.k even 4 1
975.2.bp.d 4 65.r odd 12 1
975.2.bp.d 4 195.j odd 4 1
975.2.bp.d 4 195.bf even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}$$ $$T_{5}$$ $$T_{7}^{4} + 8 T_{7}^{3} + 41 T_{7}^{2} + 130 T_{7} + 169$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$169 + 130 T + 41 T^{2} + 8 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$676 + 364 T + 50 T^{2} + 2 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$169 + 182 T + 98 T^{2} - 14 T^{3} + T^{4}$$
$37$ $$676 - 52 T + 122 T^{2} + 22 T^{3} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 3 - 3 T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$5625 + 75 T^{2} + T^{4}$$
$67$ $$169 - 416 T + 281 T^{2} - 10 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$20449 - 4862 T + 578 T^{2} - 34 T^{3} + T^{4}$$
$79$ $$( -147 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$28561 + 6422 T + 557 T^{2} + 28 T^{3} + T^{4}$$